In an exercise in my syllabus on homological algebra, I need to explicitly describe what are the $n$ isomorphism classes of extensions of $\mathbf{Z}/n\mathbf{Z}$ by $\mathbf{Z}$ for the cases $n=p$ prime, $n=pq$ with $p,q$ distinct primes and $n=4$. (The group $\operatorname{Ext}_\mathbf{Z}^1 (\mathbf{Z}/n\mathbf{Z},\mathbf{Z})$.)
For $n=p$, I succeeded in classifying these: there are $p$ short exact sequences of the form
$$0\longrightarrow \mathbf Z\stackrel{\times p}{\longrightarrow} \mathbf Z\stackrel{f}{\longrightarrow} \mathbf Z/p\mathbf Z\longrightarrow 0$$ with $f:1\mapsto \overline{a}$ with $a\in \{1,\ldots,p-1\}$ and we have the split extension
$$0\longrightarrow \mathbf Z\stackrel{}{\longrightarrow} \mathbf Z \oplus \mathbf Z/p\mathbf Z\stackrel{}{\longrightarrow} \mathbf Z/p\mathbf Z\longrightarrow 0$$ which are clearly distinct isomorphism classes.
For $n=4$, I can only find three:
$$0\longrightarrow \mathbf Z\stackrel{\times 4}{\longrightarrow} \mathbf Z\stackrel{\pi_i}{\longrightarrow} \mathbf Z/4\mathbf Z\longrightarrow 0$$ where $\pi_1:x\mapsto \overline{x}$ and $\pi_2:x\mapsto \overline{-x}$, and the split extension $$0\longrightarrow \mathbf Z\stackrel{}{\longrightarrow} \mathbf Z \oplus \mathbf Z/4\mathbf Z\stackrel{}{\longrightarrow} \mathbf Z/4\mathbf Z\longrightarrow 0.$$
For $n=pq$, we again have the split extension and we can mimic what we did for $p$ prime to obtain $(p-1)(q-1)$ non-equivalent extensions of the form
$$0\longrightarrow \mathbf Z\stackrel{\times pq}{\longrightarrow} \mathbf Z\stackrel{f}{\longrightarrow} \mathbf Z/pq\mathbf Z\longrightarrow 0$$ with $f:1\mapsto \overline{a}$ with $a\in \{1,\ldots,p-1\}\times \{1,\ldots,q-1 \}$.
Any help is appreciated!
Here's one for $n=4$: $$0\to\Bbb Z\to\Bbb Z\oplus \Bbb Z/2\Bbb Z\to\Bbb Z/4\Bbb Z\to0.$$ In the injection, $1$ goes to $(2,1)$ and in the surjection $(a,b)$ goes to $a+2b$.
In general one will get several extensions with middle term $\Bbb Z\oplus\Bbb Z/m\Bbb Z$ where $m$ is any divisor of $n$.